A superstatistical measure of distance from canonical equilibrium

Non-equilibrium systems in steady states are commonly described by generalized statistical mechanical theories such as non-extensive statistics and superstatistics. Superstatistics assumes that the inverse temperature $\beta = 1/(k_B T)$ follows some pre-established statistical distribution, however, it has been previously proved (Physica A 505, 864-870 [2018]) that $\beta$ cannot be associated to an observable function $B(\boldsymbol{\Gamma})$ of the microstates $\boldsymbol{\Gamma}$. In this work, we provide an information-theoretical interpretation of this theorem by introducing a new quantity $\mathcal{D}$, the mutual information between $\beta$ and $\boldsymbol{\Gamma}$. Our results show that $\mathcal{D}$ is also a measure of departure from canonical equilibrium, and reveal a minimum, non-zero uncertainty about $\beta$ given $\boldsymbol{\Gamma}$ for every non-canonical superstatistical ensemble. This supports the use of the mutual information as a descriptor of complexity and correlation in complex systems, also providing in some cases a sound basis for the use of Tsallis' entropic index $q$ as a measure of distance from equilibrium, being in those cases a proxy for $\mathcal{D}$.